Solution to Exercise 2: Moral Hazard

Transcription

1 olution to Exercise 2: Moral Hazard Cheng Chen chool of Economics and inance The University of Hong Kong (Cheng Chen (HKU)) Econ / 8

2 Question One: irst Best uppose the investor (i.e., principal) can observe the entrepreneur's action, and the action is veriable by the court. (Cheng Chen (HKU)) Econ / 8

3 Question One: irst Best uppose the investor (i.e., principal) can observe the entrepreneur's action, and the action is veriable by the court. In this case, these are no incentive problems, since the principal can check whether the agent chooses the assigned action. (Cheng Chen (HKU)) Econ / 8

4 Question One: irst Best uppose the investor (i.e., principal) can observe the entrepreneur's action, and the action is veriable by the court. In this case, these are no incentive problems, since the principal can check whether the agent chooses the assigned action. If the principal wants the agent to choose a l (i.e., shirking): The constraint is 0.2 2w 1/ w 1/ , (PCl) and the solution is w = w = 1 (remember the agent is risk averse) Payo to the principal is V = 0.2 (30 1) (0 1) = 5. (Cheng Chen (HKU)) Econ / 8

5 Question One: irst Best (Cont.) If the principal wants the agent to choose a h (i.e., working hard): The constraint is 0.6 2w 1/ w 1/ , (PCh) and the solution is w = w = 4 (remember the agent is risk averse) Payo to the principal is V = 0.6 (30 4) (0 4) = 14(> 5). (Cheng Chen (HKU)) Econ / 8

6 Question One: irst Best (Cont.) If the principal wants the agent to choose a h (i.e., working hard): The constraint is 0.6 2w 1/ w 1/ , (PCh) and the solution is w = w = 4 (remember the agent is risk averse) Payo to the principal is V = 0.6 (30 4) (0 4) = 14(> 5). The principal wants the agent to work hard. And the optimal contract is (w = w = 4). ince the investment cost is I = 5(< 14), it is protable to implement the project. (Cheng Chen (HKU)) Econ / 8

7 Question One: irst Best (Cont.) If the principal wants the agent to choose a h (i.e., working hard): The constraint is 0.6 2w 1/ w 1/ , (PCh) and the solution is w = w = 4 (remember the agent is risk averse) Payo to the principal is V = 0.6 (30 4) (0 4) = 14(> 5). The principal wants the agent to work hard. And the optimal contract is (w = w = 4). ince the investment cost is I = 5(< 14), it is protable to implement the project. irst Best (B) is achieved, since there is no information problem. The agent is perfectly ensured (no welfare loss due to risk aversion). B eort choice (i.e., a h ) is chosen. The project is implemented. (Cheng Chen (HKU)) Econ / 8

8 Question Two: econd Best with a h uppose the investor cannot observe the entrepreneur's action. However, the realization of the outcome is observable and veriable. (Cheng Chen (HKU)) Econ / 8

9 Question Two: econd Best with a h uppose the investor cannot observe the entrepreneur's action. However, the realization of the outcome is observable and veriable. If the investor wants the entrepreneur to choose a h, what is the optimization problem? What are the constraints? What is the optimal contract? (Cheng Chen (HKU)) Econ / 8

10 Question Two: econd Best with a h uppose the investor cannot observe the entrepreneur's action. However, the realization of the outcome is observable and veriable. If the investor wants the entrepreneur to choose a h, what is the optimization problem? What are the constraints? What is the optimal contract? Now we have two constraints (PC and IC), since the action is unobservable. The optimization problem is max 0.6(30 w ) + 0.4(0 w ) w,w 0.6 2w 1/ w 1/ , (PCh) 0.6 2w 1/ w 1/ w 1/ w 1/2 0 2, (ICh) where ICh says that the agent is worse o by choosing a h than choosing a l. (Cheng Chen (HKU)) Econ / 8

11 Question Two: econd Best with a h (Cont.) Observations: w > w ; PCh holds with equality; ICh holds with equality. The optimal contract is and 2w 1/2 2 = d h + 1 p h p h p l (d h d l ) = 4 2w 1/2 2 = d h p h p h p l (d h d l ) = 1 (Cheng Chen (HKU)) Econ / 8

12 Question Two: econd Best with a h (Cont.) Observations: w > w ; PCh holds with equality; ICh holds with equality. The optimal contract is and 2w 1/2 2 = d h + 1 p h p h p l (d h d l ) = 4 2w 1/2 2 = d h p h p h p l (d h d l ) = 1 ubstituting values of p h, p l, d h and d l into above equations, we arrive at w = 9; w = 1/4. (Cheng Chen (HKU)) Econ / 8

13 Question Two: econd Best with a h (Cont.) Observations: w > w ; PCh holds with equality; ICh holds with equality. The optimal contract is and 2w 1/2 2 = d h + 1 p h p h p l (d h d l ) = 4 2w 1/2 2 = d h p h p h p l (d h d l ) = 1 ubstituting values of p h, p l, d h and d l into above equations, we arrive at w = 9; w = 1/4. Net payo to the principal is V = 0.6 (30 9) (0 1/4) = 12.5(> 5). (Cheng Chen (HKU)) Econ / 8

14 Question Two: econd Best with a l If the investor wants the entrepreneur to choose a l, what is the optimization problem? What are the constraints? What is the optimal contract? (Cheng Chen (HKU)) Econ / 8

15 Question Two: econd Best with a l If the investor wants the entrepreneur to choose a l, what is the optimization problem? What are the constraints? What is the optimal contract? Now we have two constraints (PC and IC), since the action is unobservable. The optimization problem is max 0.2(30 w ) + 0.8(0 w ) w,w 0.2 2w 1/ w 1/2 2 0, (PCl) 0.2 2w 1/ w 1/ w 1/ w 1/2 2 2, (ICl) (Cheng Chen (HKU)) Econ / 8

16 Question Two: econd Best with a l If the investor wants the entrepreneur to choose a l, what is the optimization problem? What are the constraints? What is the optimal contract? Now we have two constraints (PC and IC), since the action is unobservable. The optimization problem is max 0.2(30 w ) + 0.8(0 w ) w,w 0.2 2w 1/ w 1/2 2 0, (PCl) 0.2 2w 1/ w 1/ w 1/ w 1/2 2 2, (ICl) Under perfect insurance (i.e., w = w ), ICl is met trivially rightarrow Optimal contract is the same as in the B case: w = w = 1; V = 0.2 (30 1) (0 1) = 5(< 12.5). (Cheng Chen (HKU)) Econ / 8

17 Question Two: Conclusion Optimal choice is to incentivize the agent to work hard and set w = 9 and w = 1/4. It is protable to implement the project, since V (a h ) = 12.5 > I = 5. Actually, if the principal incentivizes the agent to choose a l, she can only break even. (Cheng Chen (HKU)) Econ / 8

18 Question Two: Conclusion Optimal choice is to incentivize the agent to work hard and set w = 9 and w = 1/4. It is protable to implement the project, since V (a h ) = 12.5 > I = 5. Actually, if the principal incentivizes the agent to choose a l, she can only break even. Compared with the B case, the principal's payo goes down from 14 to The payment in case of has to be higher than the payment in case of L: Incentive concern. ince the agent is risk averse and earns zero payo in equilibrium, the payo to the principal is lower in the B case. (Cheng Chen (HKU)) Econ / 8

19 Question Two: Conclusion Optimal choice is to incentivize the agent to work hard and set w = 9 and w = 1/4. It is protable to implement the project, since V (a h ) = 12.5 > I = 5. Actually, if the principal incentivizes the agent to choose a l, she can only break even. Compared with the B case, the principal's payo goes down from 14 to The payment in case of has to be higher than the payment in case of L: Incentive concern. ince the agent is risk averse and earns zero payo in equilibrium, the payo to the principal is lower in the B case. If the investment cost I = 13, then the project is not protable due to the information problem, since 12.5 < 13. However, it is protable in the B case, since 14 > 13. Market is closed, and there is welfare loss. (Cheng Chen (HKU)) Econ / 8

20 Question Three: econd Best and Maximum Punishment uppose the investor cannot observe the entrepreneur's action. However, the realization of the project is observable and veriable. Now, suppose that p h becomes one now (other things unchanged). (Cheng Chen (HKU)) Econ / 8

21 Question Three: econd Best and Maximum Punishment uppose the investor cannot observe the entrepreneur's action. However, the realization of the project is observable and veriable. Now, suppose that p h becomes one now (other things unchanged). In this case, the principal can set the payment of w so low such that ICh is satised. urthermore, there is no loss for the principal, since the agent receives a xed payment when working hard (remember p h = 1 now). (Cheng Chen (HKU)) Econ / 8

22 Question Three: econd Best and Maximum Punishment uppose the investor cannot observe the entrepreneur's action. However, the realization of the project is observable and veriable. Now, suppose that p h becomes one now (other things unchanged). In this case, the principal can set the payment of w so low such that ICh is satised. urthermore, there is no loss for the principal, since the agent receives a xed payment when working hard (remember p h = 1 now). Optimal contract: w = 4: PCh is satised since p h 2 w 1/2 + (1 p h ) 2 w 1/2 d h 2 = 1 2 (4) 1/2 2 2 = 0. w = 0: ICh is met since p h 2 w 1/2 + (1 p h ) 2 w 1/2 d h 2 = 0 > p l 2 w 1/2 + (1 p l ) 2 w 1/2 d l 2 = 1.2. (Cheng Chen (HKU)) Econ / 8

23 Question Three: econd Best and Maximum Punishment uppose the investor cannot observe the entrepreneur's action. However, the realization of the project is observable and veriable. Now, suppose that p h becomes one now (other things unchanged). In this case, the principal can set the payment of w so low such that ICh is satised. urthermore, there is no loss for the principal, since the agent receives a xed payment when working hard (remember p h = 1 now). Optimal contract: w = 4: PCh is satised since p h 2 w 1/2 + (1 p h ) 2 w 1/2 d h 2 = 1 2 (4) 1/2 2 2 = 0. w = 0: ICh is met since p h 2 w 1/2 + (1 p h ) 2 w 1/2 d h 2 = 0 > p l 2 w 1/2 + (1 p l ) 2 w 1/2 d l 2 = 1.2. Maximum punishment: Realization of is possible, only when agent shirks. Therefore, principal wants to punish agent when is realized. (Cheng Chen (HKU)) Econ / 8